We have prepared the corresponding data for each criterion, turning them into fuzzy numbers.

While preparing the aptitude map with Spatial Analyst, we have discussed the fact that there were values of certain criteria which were totally ruled out: if the type of terrain is a water body or a wetland the resultant value cannot be different from zero.

The slope value is also a precluding criterion: from a certain value it becomes impossible to construct a building.

Before embarking on the elaboration of the aptitude map, we must decide how we will deal with these exclusions.

We have the choice between two approaches. The first is to do as we did when using Spatial Analyst and eliminate completely the result areas that have certain values of a criterion. The second is not to eliminate these areas and to see if all the other criteria do not “recapture” the disadvantages of the single excluding criterion.

In the first case, we will simply remove the excluded values from the input layer. In the aggregation phase of the criteria we will systematically choose the option “intersection” of the Criteria. Those areas absent in a layer will be present in all subsequent aggregations.

In the second case, a post-processing of the aptitude map will be necessary to identify the areas that exceed the value of the exclusion criterion and to study and display them separately.

In our example, we will follow the first procedure for *land occupations* and the second for the *Slope*.

**Exclusion of “water body” and “wetland” zones**

We open an editing session and, for the OccupationSolPolys layer, we enter the request:

**LANDUSE in (“Water”, “Wetlands”)**

Then we click on “Delete selection” and you save the changes.

The entities corresponding to these two values are now absent from the layer.

**Aggregation criteria**

We have to aggregate four different criteria. The Aggregate Soup command allows to aggregate two criteria. Therefore we will use it three times:

- We combine two criteria to obtain the result R1
- We add R1 to the third criterion to obtain the result R2, then
- R2 is added to the fourth criterion to obtain the final result.

**The order of the aggregation has no influence on the result.**

The command asks three questions to determine the mathematical model of the aggregation. In the case of a series of aggregations, the questions must be understood as the result of the criterion to be crossed in relation to all the other criteria.

Example:

We will start by crossing Distance to Schools and Slope. The first question asked will be:

*If the criterion FZY_distEc is ‘Very bad’ and the criterion FZY_pente is ‘Very good’,*

But you will have to understand:

*If the criterion FZY_distEc is ‘Very bad’ and THE ENTIRE OTHER CRITERIA is the criterion ‘Very good’*

The other question that must be asked systematically is the opposite question:

*If ALL OTHER CRITERIA is ‘Very bad’ and the criterion FZY_distEc is ‘Very good’ will I give the same answer?*

As we explained in the theoretical basis of fuzzy commands, this aggregation command works by assuming the symmetry of the responses. Whether you put a criterion in the right or left list of the command, it should not change the answers for the three questions.

If this changes the answer, you are in the special case of the criteria of unequal importance and you should not use the flexible aggregation command presented so far. We will discuss another experimental command to deal with these cases. For the moment, we will consider that, in our example, the answer is symmetrical.

**Aggregation of Distances to schools and recreational centres**

We will use the *Soft Aggregation* command to make the intersection of the two DistanceSchools and DistanceRecre layers:

We select in each layer the fuzzy criterion that we created in the previous articles while checking that the type of operation is “Intersection”.

The difference with the procedure followed with Spatial Analyst is that here we ask the questions as we usually ask them and the answers are as we usually do. It’s clear that according to our role as actor in the project, the answers we will deliver will not be the same: as a pupil’s parent we will not answer the same thing a promoter of the building will, or a local elected representative. Therefore, the final aptitude map will not be the same according to our answers, and this is what happens regularly in all development projects: each actor has his own “cartography” of feasible sites. Unlike the treatment with Spatial Analyst, here we will have a record of the responses, and, ultimately, we can see where the opinions of different players differ, ahead the final result. We will then be able to return to the real source of disagreement without limiting ourselves to a statement of the different final results.

Here, for our example, we offer a classical answer: medium – medium – good which is an average of the two criteria. In another case, we can consider that the non-satisfaction of a criterion is a handicap for the result and we will note the result lower than an average result, or the criteria is irrelevant or even the non-satisfaction of the criteria do not compromise the result. Then, we will come up with a result higher than the average.

We execute the command and obtain the first aggregation result:

**Aggregation of the first result and land use**

We will use the Soft Aggregation command to intersect the two Agreg_Eco_Rec and OccupationSolPolys layers:

We execute the command and get the second aggregation result:

**Aggregation of the second** ** ** **result and slope of the terrain**

We will use the Soft Aggregation command to intersect the two Agreg_Eco_Rec_Occ and PentePolys layers:

We execute the command and get the third and last aggregation result:

The following image shows the two aptitude maps we built, on the left according the Spatial Analyst tutorial and on the right according the fuzzy aggregation method:

Among the innumerable differences that we can see between the two methods:

By the classical method, we have no results for the satisfaction values between 1, 2, 3 and 10. The main results are concentrated in an overall satisfaction index between 4 and 8. This is an artefact that we ourselves have created using the weighting system. That is, we will propose sites that fit a mediocre-good range.

The result of the fuzzy aggregation ranges from 0 to 1 (which corresponds to 0 -10 of the classical method). Contrary to what one might think, the fuzzy method is more restrictive than the classical: by defining the fuzzy numbers we have established true boundaries (0 and 1) compared to the linear reclassification method between 0 and 10 of Spatial Analyst.

In the next article we will discuss in more detail the difference among the results and, especially, the consideration of the criteria of unequal importance.